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Find the Volume of Each Pyramid. 1. SOLUTION

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Find the Volume of Each Pyramid. 1. SOLUTION 12-5 Volumes of Pyramids and Cones Find the volume of each pyramid. 1. SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a right triangle with legs of 9 inches and 5 inches and the height of the pyramid is 10 inches. ANSWER: 3 75 in 2. SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a regular pentagon with sides of 4.4 centimeters and an apothem of 3 centimeters. The height of the pyramid is 12 centimeters. eSolutions Manual - Powered by Cognero Page 1 ANSWER: 132 cm3 3. a rectangular pyramid with a height of 5.2 meters and a base 8 meters by 4.5 meters SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a rectangle with a length of 8 meters and a width of 4.5 meters. The height of the pyramid is 5.2 meters. ANSWER: 3 62.4 m 4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters. ANSWER: 298.7 m3 Find the volume of each cone. Round to the nearest tenth. 5. SOLUTION: The volume of a circular cone is , or , where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter of this cone is 7 inches, the radius is or 3.5 inches. The height of the cone is 4 inches. ANSWER: 3 51.3 in 6. SOLUTION: Use trigonometry to find the radius r. The volume of a circular cone is , or , where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters. ANSWER: 168.1 cm3 7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters SOLUTION: The volume of a circular cone is , or , where B is the area of the base, h is the height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters. ANSWER: 3 28.1 mm 8. a cone with a slant height of 25 meters and a radius of 15 meters SOLUTION: Use the Pythagorean Theorem to find the height h of the cone. Then find its volume. So, the height of the cone is 20 meters. ANSWER: 4712.4 m3 9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have to accommodate. Round to the nearest tenth. SOLUTION: The volume of a circular cone is , where B is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet. ANSWER: 3 513,333.3 ft CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary. 10. SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. ANSWER: 605 in3 11. SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. ANSWER: 3 105.8 mm 12. SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. ANSWER: 482.1 m3 13. SOLUTION: The volume of a pyramid is , where B is the base and h is the height of the pyramid. The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle. The apothem is . ANSWER: 3 233.8 cm 14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. ANSWER: 1376.7 ft3 15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters SOLUTION: Find the height of the right triangle. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. ANSWER: 3 35.6 cm 16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. SOLUTION: The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a of the right triangle and then find the area of the triangle. The length of the other leg of the right triangle is 6 cm. So, the area of the base B is 24 cm2. Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h. Therefore, the height of the triangular pyramid is 18 cm. ANSWER: 18 cm Find the volume of each cone. Round to the nearest tenth. 17. SOLUTION: The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches. 3 Therefore, the volume of the cone is about 235.6 in . ANSWER: 235.6 in3 18. SOLUTION: The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters. Therefore, the volume of the cone is about 134.8 3 cm . ANSWER: 134.8 cm3 19. SOLUTION: Use a trigonometric ratio to find the height h of the cone. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters. Therefore, the volume of the cone is about 1473.1 3 cm . ANSWER: 1473.1 cm3 20. SOLUTION: Use trigonometric ratios to find the height h and the radius r of the cone. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. 3 Therefore, the volume of the cone is about 2.8 ft . ANSWER: 2.8 ft3 21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches SOLUTION: The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches. Therefore, the volume of the cone is about 1072.3 3 in . ANSWER: 1072.3 in3 22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter SOLUTION: The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean Theorem to find the height h of the cone. 3 Therefore, the volume of the cone is about 5.8 cm . ANSWER: 5.8 cm3 23. SNACKS Approximately how many cubic centimeters of roasted peanuts will completely fill a paper cone that is 14 centimeters high and has a base diameter of 8 centimeters? Round to the nearest tenth. SOLUTION: The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8 centimeters, the radius is or 4 centimeters. The height of the cone is 14 centimeters. 3 Therefore, the paper cone will hold about 234.6 cm of roasted peanuts. ANSWER: 234.6 cm3 24. CCSS MODELING The Pyramid Arena in Memphis, Tennessee, is the third largest pyramid in the world. It is approximately 350 feet tall, and its square base is 600 feet wide. Find the volume of this pyramid. SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. ANSWER: 3 42,000,000 ft 25. GARDENING The greenhouse is a regular octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the greenhouse? SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.
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